The approximation to the solution of partial differential equations (PDEs) can be substantially improved by employing smooth basis functions. The recently introduced mollified basis functions are constructed through mollification, or convolution, of piecewise polynomials defined over a cell with a smooth mollifier. The approximation properties of the mollified basis functions are governed by the order of the piecewise polynomial function and the smoothness of the mollifier. Importantly, these approximation and smoothness properties do not degrade over extraordinary vertices, making it well-suited for polytopic partitions, such as Voronoi tessellation. In this work, we exploit such high-order and high-smoothness characteristics of the mollified basis functions for solving PDEs using the point collocation method (PCM). The basis functions are evaluated at a set of collocation points within the domain. Additionally, boundary conditions are evaluated at specified collocation points distributed on the domain boundaries. To ensure the stability of the linear system, it is imperative that the number of collocation points exceeds the total number of basis functions. Consequently, the resulting linear system is overdetermined and is solved using the least square technique. Furthermore, appropriate scaling factors are applied to improve the conditioning of the matrix system due to the use of high-order polynomial basis and their derivatives. The presented numerical examples confirm the optimal convergence of the proposed approximation scheme for Poisson, linear elasticity, and biharmonic problems. The presented study also investigates the influence of the mollifier and the spatial distribution of collocation points.